Connected component (graph theory) in the context of Minimum spanning tree

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network.

In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex ${\displaystyle s}$ can reach a vertex ${\displaystyle t}$ (and ${\displaystyle t}$ is reachable from ${\displaystyle s}$) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with ${\displaystyle s}$ and ends with ${\displaystyle t}$.

In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph. Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric (${\displaystyle s}$ reaches ${\displaystyle t}$ iff ${\displaystyle t}$ reaches ${\displaystyle s}$). The connected components of an undirected graph can be identified in linear time. The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph (which, incidentally, need not be symmetric).